Dirac operators on cobordisms: Degenerations and surgery

Abstract

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator Dt on each regular level set Ct of a fixed Morse function defining this cobordism. We show that as we approach the critical level set C0 from above and from below these operators converge in the gap topology to (different) selfadjoint operators D± that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators Dt on the complement of C0 and the Kashiwara-Wall index of a triplet of finite dimensional lagrangian spaces canonically determined by C0. CONTENTS Introduction 1 1. A model degeneration 4 2. The Atiyah-Patodi-Singer theorem 12 3. Dolbeault operators on two-dimensional cobordisms 18 4. The Kashiwara-Wall index 26 References 31